22 



CONCRETE REPRESENTATIONS OF 



on the circle, i.e. the cross-ratio of the pencil 0(PQ, XY) 

 where is any point on the circle. 



17. The expression for the line-element can now be found 

 by making PQ infinitesimal. 



We have, by Ptolemy's Theorem, 



PX . Q Y=PQ . X Y+PY . QX. 



Hence 



cfc^log (l + 



Let OP (Fig. 1) cut the circle PXY again in R and the fixed 

 circle in A, B. Then R is a fixed point so that PR is constant. 



Also 



= -=& fixed ratio=e, 



PX PY 

 and PR.XY=PX.RY+PY.RX=2e.PX.PY. 



Therefore -&-* T>V=J^ an( * * s therefore a function of the 

 tr JL . JT I rti 



position of P alone. 



FIG. 1 



To find its value we may take any orthogonal circle through 

 P, say the straight line PR. 



XY AB 



Then 



Hence 



PX.PY~PA. 



O.. . / J, 



d*- 



